🪡K-Theory Unit 2 – K-Theory: Grothendieck Group and K(X) Definition

Key Concepts and Definitions

• Grothendieck group is a construction that turns a commutative monoid into an abelian group, allowing for subtraction and division operations
• K-theory studies vector bundles on topological spaces and the Grothendieck group of vector bundles, denoted as K(X)
• Vector bundles are a collection of vector spaces parametrized by points in a topological space X
  • Consist of a total space E, a base space X, and a projection map $\pi: E \to X$
  • Each fiber $\pi^{-1}(x)$ for $x \in X$ is a vector space
• Isomorphism of vector bundles is a homeomorphism between total spaces that preserves the vector space structure of fibers
• Whitney sum $\oplus$ is an operation that combines two vector bundles over the same base space into a new vector bundle
  • Fibers of the Whitney sum are direct sums of corresponding fibers from the original bundles
• Tensor product $\otimes$ is another operation on vector bundles that multiplies fibers pointwise using the tensor product of vector spaces

Historical Context and Development

• K-theory originated in the late 1950s and early 1960s through the work of Alexander Grothendieck and Michael Atiyah
• Grothendieck introduced the idea of studying vector bundles using the Grothendieck group construction in his proof of the Grothendieck-Riemann-Roch theorem
• Atiyah and Hirzebruch further developed K-theory and introduced the notion of K(X) for a topological space X
• K-theory became a powerful tool in algebraic geometry, topology, and other areas of mathematics
• Quillen introduced higher algebraic K-theory in the 1970s, generalizing the ideas of Grothendieck and Atiyah-Hirzebruch
• Connections between K-theory and other areas like operator algebras and physics have been explored since its inception

Grothendieck Group Construction

• Start with a commutative monoid M, which is a set with an associative and commutative binary operation + and an identity element 0
• Consider the set of formal differences $M \times M$ and define an equivalence relation $\sim$ on this set
  • $(a, b) \sim (c, d)$ if and only if there exists $e \in M$ such that $a + d + e = b + c + e$
• The Grothendieck group of M, denoted $G(M)$, is the set of equivalence classes $[(a, b)]$ under $\sim$
• Define addition on $G(M)$ by $[(a, b)] + [(c, d)] = [(a + c, b + d)]$
• The identity element in $G(M)$ is $[(0, 0)]$, and the inverse of $[(a, b)]$ is $[(b, a)]$
• The Grothendieck group construction turns the commutative monoid M into an abelian group $G(M)$

Properties of the Grothendieck Group

• The Grothendieck group G(M) satisfies the universal property for abelian groups
  • For any abelian group A and monoid homomorphism $f: M \to A$, there exists a unique group homomorphism $\tilde{f}: G(M) \to A$ such that $\tilde{f} \circ \iota = f$, where $\iota: M \to G(M)$ is the natural inclusion
• The Grothendieck group construction is functorial
  • A monoid homomorphism $f: M \to N$ induces a group homomorphism $G(f): G(M) \to G(N)$
• The Grothendieck group of a product of monoids is isomorphic to the product of their Grothendieck groups
  • $G(M \times N) \cong G(M) \times G(N)$
• The Grothendieck group of a free commutative monoid on a set X is isomorphic to the free abelian group on X

K(X) Definition and Significance

• For a compact Hausdorff space X, K(X) is defined as the Grothendieck group of the monoid of isomorphism classes of vector bundles over X under the Whitney sum operation
• Elements of K(X) are formal differences of vector bundles, i.e., $[E] - [F]$, where E and F are vector bundles over X
• K(X) is an abelian group with addition given by the Whitney sum and the inverse of $[E]$ being $[-E]$
• The tensor product of vector bundles induces a ring structure on K(X), making it a commutative ring
• K(X) is a contravariant functor from the category of compact Hausdorff spaces to the category of commutative rings
  • A continuous map $f: X \to Y$ induces a ring homomorphism $f^*: K(Y) \to K(X)$ by pulling back vector bundles
• K(X) is a generalized cohomology theory, satisfying the Eilenberg-Steenrod axioms except for the dimension axiom
• The Chern character provides a ring homomorphism from K(X) to the even-dimensional cohomology ring of X with rational coefficients

Applications in Mathematics

• K-theory has applications in various areas of mathematics, including algebraic geometry, topology, number theory, and operator algebras
• In algebraic geometry, K-theory is used to study algebraic vector bundles and coherent sheaves on schemes
  • The Grothendieck-Riemann-Roch theorem relates the Chern character of a coherent sheaf to its pushforward under a proper morphism
• In topology, K-theory is used to study vector bundles on manifolds and CW complexes
  • The Atiyah-Singer index theorem relates the index of an elliptic operator to topological invariants of the underlying manifold
• In number theory, algebraic K-theory is used to study the K-groups of rings and schemes
  • The Quillen-Lichtenbaum conjecture relates the algebraic K-theory of a number field to its étale cohomology
• In operator algebras, K-theory is used to study projections and unitaries in C*-algebras and von Neumann algebras
  • The Connes-Kasparov conjecture relates the K-theory of certain C*-algebras to their cyclic cohomology

Connections to Other Algebraic Theories

• K-theory has connections to various other algebraic theories, including cohomology theories, representation theory, and category theory
• K-theory is a generalized cohomology theory, and there are natural transformations between K-theory and other cohomology theories like singular cohomology and de Rham cohomology
• The representation ring of a compact Lie group G, denoted R(G), is isomorphic to the K-theory of the classifying space BG
  • This connection allows for the study of representations using topological methods
• K-theory can be formulated in the language of category theory using the notion of a Waldhausen category
  • The K-theory of a Waldhausen category satisfies certain universal properties and can be computed using the Q-construction
• The Baum-Connes conjecture relates the K-theory of the reduced C*-algebra of a discrete group to the equivariant K-homology of its classifying space
  • This conjecture has important implications in geometry, topology, and representation theory

Exercises and Problem-Solving Techniques

• To compute K(X) for a given space X, one can use various techniques depending on the structure of X
  • For contractible spaces, K(X) is isomorphic to the integers Z
  • For disjoint unions, $K(X \sqcup Y) \cong K(X) \oplus K(Y)$
  • For product spaces, $K(X \times Y) \cong K(X) \otimes K(Y)$
• The Mayer-Vietoris sequence is a powerful tool for computing K(X) when X can be decomposed into simpler pieces
  • For an open cover $X = U \cup V$, there is a long exact sequence relating K(X) to K(U), K(V), and $K(U \cap V)$
• The Atiyah-Hirzebruch spectral sequence relates the K-theory of a space X to its singular cohomology
  • The $E_2$ page of the spectral sequence is given by $E_2^{p,q} = H^p(X; K^q(point))$, and the sequence converges to $K^*(X)$
• To show that two vector bundles are isomorphic, one can use techniques like the clutching construction or the transition functions approach
• Computing the Chern character of a vector bundle involves understanding its Chern classes and using the splitting principle
• Practice problems can include computing K(X) for various spaces (spheres, projective spaces, tori), proving isomorphisms between vector bundles, and applying the Grothendieck-Riemann-Roch theorem or the Atiyah-Singer index theorem in specific cases